Lecture Notes of
Seminario Interdisciplinare di Matematica
Vol. 14(2017), pp. 1 – 15.
2–Factors of regular graphs: an updated survey
Marien Abreu1 and Domenico Labbate2
Abstract. A 2–factor of a graph G is a 2–regular spanning subgraph of G. We give
an updated survey on results on the structure of 2–factors in regular graphs obtained in
the last years by several authors.
1. Introduction
All graphs considered are finite and simple (without loops or multiple edges). We
shall use the term multigraph when multiple edges are permitted. For definitions
and notations not explicitly stated the reader may refer to Bondy and Murty’s book
[13].
Several authors have considered the number of Hamilton circuits in k–regular
graphs and there are interesting and beautiful results and conjectures in the literature. In particular, C.A.B. Smith (1940, cf. Tutte [45]) proved that each edge of a
3–regular multigraph lies in an even number of Hamilton circuits. This result was
extended to multigraphs in which each vertex has odd degree by Thomason [42].
A multigraph with exactly one Hamilton circuit is said to be uniquely hamiltonian. Thomason’s result implies that there are no regular uniquely hamiltonian
multigraphs of odd degree. In 1975, Sheehan [40] posed the following famous conjecture:
Conjecture 1.1. There are no uniquely hamiltonian k–regular graphs for all integers k 3.
It is well known that it is enough to prove it for k = 4. This conjecture has been
verified by Thomassen for bipartite graphs, [43] (under the weaker hypothesis that
G has minimum degree 3), and for k–regular graphs when k 300, [44]. This value
has been improved by Ghandehari and Hatami for k
48 [21] and, recently, by
Haxell, Seamone and Verstraete [24] for k > 22.
In this context, several recent papers addressed the problem of characterizing
families of graphs (particularly regular graphs) which have certain conditions imposed on their 2–factors. In this survey we present the main results obtained in the
1M. Abreu, Università degli Studi della Basilicata, Dipartimento di Matematica, Informatica
ed Economia, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy; [email protected]
2D. Labbate, Università degli Studi della Basilicata, Dipartimento di Matematica, Informatica
ed Economia, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy; [email protected]
This research was carried out within the activity of I.N.d.A.M.-G.N.S.A.G.A. and supported
by the Italian Ministry M.I.U.R.
Keywords. 2-factors, snarks, cycles.
AMS Subject Classification. 05C70, 05C75, 05C35, 05C45.
2
Marien Abreu and Domenico Labbate
last fifteen years and we give an update of the main open problems in this field.
We will also discuss relations of these problems with a particular class of snarks i.e.
the odd 2–factored snarks.
2. Preliminaries
An r–factor of a graph G is an r–regular spanning subgraph of G. Thus a 2–
factor of a graph G is a 2–regular spanning subgraph of G. A 1–factorization of G
is a partition of the edge set of G into 1–factors.
Let G be a bipartite graph with bipartition (X, Y ) such that |X| = |Y |, and A
be its adjacency matrix. In general 0  |det(A)|  per(A). We say that G is det–
extremal if |det(A)| = per(A). Let X = {x1 , x2 , · · · , xn } and Y = {y1 , y2 , · · · , yn }
be the bipartition of G. For F a 1–factor of G define the sign of F , sgn(F ), to
be the sign of the permutation of {1, 2, · · · , n} corresponding to F . (Thus G is
det–extremal if and only if all 1–factors of G have the same sign.) The following
elementary result is a special case of [32, Lemma 8.3.1].
Lemma 2.1. Let F1 , F2 be 1-factors in a bipartite graph G and t be the number of
circuits in F1 [F2 of length congruent to zero modulo four. Then sgn(F1 )sgn(F2 ) =
( 1)t .
Let G, G1 , G2 be graphs such that G1 \ G2 = ;. Let y 2 V (G1 ) and x 2 V (G2 )
such that dG1 (y) = 3 = dG2 (x). Let x1 , x2 , x3 be the neighbours of y in G1 and
y1 , y2 , y3 be the neighbours of x in G2 . If G = (G1 y)[(G2 x)[{x1 y1 , x2 y2 , x3 y3 },
then we say that G is a star product of G1 and G2 and write G = (G1 , y) ⇤ (G2 , x).
The Heawood graph H0 is the bipartite graph associated with the point/ line
incidence matrix of the Fano plane P G(2, 2). Let H be the class of graphs obtained
from the Heawood graph by repeated star products.
These graphs were used by McCuaig in [35] to characterise the 3–connected cubic
det-extremal bipartite graphs:
Theorem 2.2 ([35]). A 3-connected cubic bipartite graph is det-extremal if and
only if it belongs to H.
Note (i) Theorem 2.2 has been improved for connectivity 2 graphs by Funk, Jackson, Labbate and Sheehan in [17]
(ii) Bipartite graphs G with the more general property that some of the entries
in the adjacency matrix A of G can be changed from 1 to 1 in such a way that
the resulting matrix A⇤ satisfies per(A) = det(A⇤ ) have been characterised in [31,
34, 36].
3. 2–factor hamiltonian graphs
A graph with a 2–factor is said to be 2–factor hamiltonian if all its 2–factors
are Hamilton cycles. Examples of such graphs are K4 , K5 , K3,3 , the Heawood
graph H0 , and the cubic graph of girth five obtained from a 9-circuit by adding
three vertices, each joined to three vertices of the 9-circuit. (The latter graph is
the ‘Triplex graph’ of Robertson, Seymour and Thomas.)
The following property is easy to prove and, at the same time, important for
approaching a characterization of this family of graphs.
2–Factors of regular graphs: an updated survey
3
Proposition 3.1 ([18]). If a bipartite graph G can be represented as a star product
G = (G1 , y) ⇤ (G2 , x), then G is 2–factor hamiltonian if and only if G1 and G2 are
2–factor hamiltonian.
Note that K4 ⇤ K4 shows that the star products of non–bipartite 2–factor hamiltonian graphs is not necessarily 2–factor hamiltonian.
Using Proposition 3.1, Funk, Jackson, Labbate and Sheehan constructed an infinite family of 2–factor hamiltonian cubic bipartite graphs by taking iterated star
products of K3,3 and H0 [18]. They conjecture that these are the only non-trivial
2–factor hamiltonian regular bipartite graphs.
Conjecture 3.2 ([18]). Let G be a 2–factor hamiltonian k-regular bipartite graph.
Then either k = 2 and G is a circuit or k = 3 and G can be obtained from K3,3
and H0 by repeated star products.
If proved, Conjecture 3.2 will allow to completely characterize the family of 2–
factor hamiltonian regular bipartite graphs. In the 80’s Sheehan posed the following
Conjecture 3.3. There are no 2–factor hamiltonian k–regular bipartite graphs for
all integers k 4.
The following properties have been proved by Labbate [28, 29] for an equivalent
family of cubic graphs (cf. subsection 3.1), and then by Funk, Jackson, Labbate
and Sheehan [18] for 2–factor hamiltonian graphs:
Lemma 3.4 ([29, 28, 18]). Let G be a 2–factor hamiltonian cubic bipartite graph.
Then G is 3–connected and |V (G)| ⌘ 2 (mod 4).
A graph H is ‘maximally’ 2–factor hamiltonian if the multigraph G obtained by
adding an edge e with endvertices u, v to H has a disconnected 2–factor containing
e.
Lemma 3.5 ([18, Lemma 3.4 (a)(i)]). Graphs obtained by taking star products of
H0 are maximally 2–factor hamiltonian.
Funk, Jackson, Labbate and Sheehan in [18] proved Conjecture 3.3 applying
Lemmas 2.1, 3.4, 3.5 and Theorem 2.2:
Theorem 3.6 ([18]). Let G be a 2–factor hamiltonian k–regular bipartite graph.
Then k  3.
Theorem 3.6 has inspired further results by Faudree, Gould and Jacobsen [15]
that determined the maximum number of edges in both 2–factor hamiltonian graphs
and 2–factor hamiltonian bipartite graphs. In particular they proved the following:
Theorem 3.7 ([15]). If G is a bipartite 2–factor hamiltonian graph of order n then
( 2
n /8 + n/2
if n ⌘ 0 mod 4 ,
|E(G)| 
2
n /8 + n/2 + 1/2 if n ⌘ 2 mod 4 ,
and the bound is sharp.
Theorem 3.8 ([15]). If G is a 2–factor hamiltonian graph of order n then
|E(G)|  dn2 /4 + n/4e
and the bound is sharp for all n
6.
4
Marien Abreu and Domenico Labbate
In addition, Diwan [14] has shown that
Theorem 3.9. K4 is the only 3–regular 2–factor hamiltonian planar graph.
Conjecture 3.2 has been partially solved in terms of minimally 1–factorable cubic
bipartite graphs as we explain in the following subsection.
3.1. Minimally 1-factorable graphs. Let G be a k–regular bipartite graph. We
say that G is minimally 1–factorable if every 1–factor of G is contained in a unique
1–factorization of G.
The results cited above by Funk, Jackson, Labbate and Sheehan were inspired
by results on minimally 1-factorable graphs obtained in [19, 28, 29, 30]. It can be
seen that:
Proposition 3.10 ([18]). If G is minimally 1–factorable then G is 2–factor hamiltonian. If k = 2, 3, then G is minimally 1–factorable if and only if G is 2–factor
hamiltonian.
Theorem 3.6 extends the result of [19] that minimally 1–factorable k–regular
bipartite graphs exist only when k  3.
Furthermore, Labbate in [30] proved the following characterization:
Theorem 3.11 ([30]). Let G be a minimally 1–factorable k–regular bipartite graph
of girth 4. Then either k = 2 and G is a circuit or k = 3 and G can be obtained
from K3,3 by repeated star products.
Hence, it follows from results in [29] that a smallest counterexample to Conjecture 3.2 is cubic and cyclically 4-edge connected, and from Theorem 3.11 that it has
girth at least six. Thus, to prove the conjecture, it would suffice to show that the
Heawood graph is the only 2-factor hamiltonian cyclically 4–edge connected cubic
bipartite graph of girth at least six.
This seems a very difficult task to achieve at least with the techniques used so
far. We have obtained, jointly with Sheehan in [5], partial results using irreducible
Levi graphs (cf. Section 5.1 and Theorem 5.10).
4. 2–factor isomorphic graphs
The family of 2–factor hamiltonian k–regular graphs can be extended to the
family of connected k–regular graphs with the more general property that all their
2–factors are isomorphic, i.e. the family of 2–factor isomorphic k–regular bipartite
graph.
Examples of such graphs are given by all the 2–factor hamiltonian and the Petersen graph (which is 2–factor isomorphic since it has all its 2–factors of length
(5, 5) but it is not 2-factor hamiltonian). Note that star product preserves also
2–factor isomorphic regular graphs.
In [7] Aldred, Funk, Jackson, Labbate and Sheehan proved the following existence theorem
Theorem 4.1 ([7]). Let G be a 2–factor isomorphic k–regular bipartite graph. Then
k  3.
They also conjecture that the family of 2–factor isomorphic and the one of 2–
factor hamiltonian k–regular bipartite graphs are, in fact, the same.
2–Factors of regular graphs: an updated survey
5
Conjecture 4.2 ([7]). Let G be a connected k-regular bipartite graph. Then G is
2–factor isomorphic if and only if G is 2–factor hamiltonian.
Abreu, Diwan, Jackson, Labbate and Sheehan proved in [3] that Conjecture 4.2
is false applying the following construction:
Proposition 4.3 ([3]). Let Gi be a 2-factor hamiltonian cubic bipartite graph with
k vertices and ei = ui vi 2 E(Gi ) for i = 1, 2, 3. Let G be the graph obtained from
the disjoint union of the graphs Gi ei by adding two new vertices w and z and new
edges wui and zvi for i = 1, 2, 3. Then G is a non-hamiltonian connected 2-factor
isomorphic cubic bipartite graph of edge-connectivity two.
Given a set {G1 , G2 , · · · , Gk } of 3–edge–connected cubic bipartite graphs let
SP(G1 , G2 , · · · , Gk ) be the set of cubic bipartite graphs which can be obtained
from G1 , G2 , · · · , Gk by repeated star products. In Section 3 we have seen that
it was shown in [18] that all graphs in SP(K3,3 , H0 ) are 2–factor hamiltonian.
Thus we may apply Proposition 4.3 by taking G1 = G2 = G3 to be any graph
in SP(K3,3 , H0 ) to obtain an infinite family of 2–edge–connected non–hamiltonian
2–factor isomorphic cubic bipartite graphs. This family gives counterexamples to
the Conjecture 4.2. Note, however, that Conjecture 4.2 can be modified as follows:
Conjecture 4.4 ([3]). Let G be a 3–edge–connected 2–factor isomorphic cubic
bipartite graph. Then G is a 2–factor hamiltonian cubic bipartite graph.
In [1, 2] Abreu, Aldred, Funk, Jackson and Sheehan proved existence theorems
also for the directed and non–bipartite graphs case as follows:
For v a vertex of a digraph D, let d+ (v) and d (v) denote the out–degree and
in–degree of v. We say that D is k–diregular if for all vertices v of G, we have
d+ (v) = d (v) = k.
Theorem 4.5 ([1, 2]). Let D be a digraph with n vertices and X be a directed
2-factor of D. Suppose that either
(a) d+ (v)
+
blog2 nc + 2 for all v 2 V (D), or
(b) d (v) = d (v)
4 for all v 2 V (D).
Then D has a directed 2-factor Y with Y ⇠
6 X.
=
Corollary 4.6 ([1]). Let G be a k–diregular directed graph. Then k  3.
Theorem 4.7 ([1, 2]). Let G be a graph with n vertices and X be a 2-factor of G.
Suppose that either
(a) d(v)
2(blog2 nc + 2) for all v 2 V (G), or
(b) G is a 2k-regular graph for some k
4.
Then G has a 2-factor Y with Y ⇠
6 X.
=
They have also posed the following open problems and conjecture:
Question 4.8 ([2]). Do there exist 2–factor isomorphic bipartite graphs of arbitrarily large minimum degree?
Question 4.9 ([2]). Do there exist 2–factor isomorphic regular graphs of arbitrarily
large degree?
6
Marien Abreu and Domenico Labbate
Conjecture 4.10 ([1]). The graph K5 is the only 2–factor hamiltonian 4–regular
non–bipartite graph.
5. Pseudo 2–factor isomorphic graphs
In [3] Abreu, Diwan, Jackson, Labbate and Sheehan extended the above mentioned results on regular 2–factor isomorphic bipartite graphs to the more general
family of pseudo 2–factor isomorphic graphs i.e. graphs G with the property that
the parity of the number of circuits in a 2–factor is the same for all 2–factors of G.
Examples of such graphs are given by all the 2–factor isomorphic regular graphs
and the Pappus graph (i.e. the point/line incidence graph of the Pappus configuration). The family of pseudo 2–factor isomorphic is wider than the one of 2–factor
isomorphic regular bipartite graphs:
Proposition 5.1 ([3]). The Pappus graph P0 is pseudo 2–factor isomorphic but
not 2–factor isomorphic.
In [3] Abreu, Diwan, Jackson, Labbate and Sheehan proved the following existence theorem:
Theorem 5.2 ([3]). Let G be a pseudo 2–factor isomorphic k–regular bipartite
graph. Then k 2 {2, 3}.
They have also shown that there are no planar pseudo 2–factor isomorphic cubic
bipartite graphs.
Theorem 5.3 ([3]). Let G be a pseudo 2–factor isomorphic cubic bipartite graph.
Then G is non–planar.
Star products preserve also the property of being pseudo 2–factor isomorphic in
the family of cubic bipartite graphs.
Lemma 5.4 ([3]). Let G be a star product of two pseudo 2–factor isomorphic cubic
bipartite graphs G1 and G2 . Then G is also pseudo 2–factor isomorphic.
Thus K3,3 , H0 and P0 can be used to construct an infinite family of 3–edge–
connected pseudo 2–factor isomorphic cubic bipartite graphs.
Given a set {G1 , G2 , · · · , Gk } of 3–edge–connected cubic bipartite graphs let
SP(G1 , G2 , · · · , Gk ) be the set of cubic bipartite graphs which can be obtained
from G1 , G2 , · · · , Gk by repeated star products. Lemma 5.4 implies that all graphs
in SP(K3,3 , H0 , P0 ) are pseudo 2–factor isomorphic. In [3] Abreu, Diwan, Jackson,
Labbate and Sheehan conjectured that these are the only 3–edge–connected pseudo
2–factor isomorphic cubic bipartite graphs.
Conjecture 5.5 ([3]). Let G be a 3–edge–connected cubic bipartite graph. Then G
is pseudo 2–factor isomorphic if and only if G belongs to SP(K3,3 , H0 , P0 ).
Recall that McCuaig [35] has shown that a 3–edge–connected cubic bipartite
graph G is det-extremal if and only if G 2 SP(H0 ).
Let G be a graph and E1 be an edge-cut of G. We say that E1 is a non-trivial
edge-cut if all components of G E1 have at least two vertices. The graph G is
essentially 4–edge–connected if G is 3–edge–connected and has no non-trivial 3–
edge–cuts. Let G be a cubic bipartite graph with bipartition (X, Y ) and K be a
non-trivial 3–edge–cut of G. Let H1 , H2 be the components of G K. We have
seen that G can be expressed as a star product G = (G1 , yK ) ⇤ (G2 , xK ) where
2–Factors of regular graphs: an updated survey
7
G1 yK = H1 and G2 xK = H2 . We say that yK , respectively xK , is the
marker vertex of G1 , respectively G2 , corresponding to the cut K. Each non-trivial
3–edge–cut of G distinct from K is a non-trivial 3–edge–cut of G1 or G2 , and vice
versa. If Gi is not essentially 4–edge–connected for i = 1, 2, then we may reduce
Gi along another non-trivial 3-edge-cut. We can continue this process until all the
graphs we obtain are essentially 4–edge–connected. We call these resulting graphs
the constituents of G. It is easy to see that the constituents of G are unique i.e.
they are independent of the order we choose to reduce the non-trivial 3–edge–cuts
of G.
It is also easy to see that Conjecture 5.5 holds if and only if Conjectures 5.6 and
5.7 below are both valid.
Conjecture 5.6 ([3]). Let G be an essentially 4–edge–connected pseudo 2–factor
isomorphic cubic bipartite graph. Then G 2 {K3,3 , H0 , P0 }.
Conjecture 5.7 ([3]). Let G be a 3–edge–connected pseudo 2–factor isomorphic
cubic bipartite graph and suppose that G = G1 ⇤ G2 . Then G1 and G2 are both
pseudo 2–factor isomorphic.
In [3] Abreu, Diwan, Jackson, Labbate and Sheehan obtained partial results on
Conjectures 5.6 and 5.7 as follows:
Theorem 5.8 ([3]). Let G be an essentially 4–edge–connected pseudo 2–factor
isomorphic cubic bipartite graph. Suppose G contains a 4–circuit. Then G = K3,3 .
They used Theorem 5.8 to deduce some evidence in favour Conjecture 5.5.
Theorem 5.9 ([3]). Let G be a 3–edge-connected pseudo 2–factor isomorphic bipartite graph. Suppose G contains a 4–cycle C. Then C is contained in a constituent
of G which is isomorphic to K3,3 .
Note that Theorem 5.9 generalizes Theorem 3.11 obtained by Labbate in [30] for
minimally 1–factorable bipartite cubic graphs (or equivalently 2–factor hamiltonian
cubic bipartite graphs) to the family of pseudo 2–factor isomorphic bipartite graph.
Furthermore, Theorem 5.9 leaves the characterization of pseudo 2–factor isomorphic
bipartite graph open for girth 6.
Recently, Abreu, Labbate and Sheehan [6] gave a partial solution to this open
case in terms of irreducible configuration of Levi graphs as described in the next
subsection.
5.1. Irreducible pseudo 2-factor isomorphic cubic bipartite graphs. An
incidence structure is linear if two di↵erent points are incident with at most one line.
A symmetric configuration nk (or nk configuration) is a linear incidence structure
consisting of n points and n lines such that each point and line is respectively
incident with k lines and points. Let C be a symmetric configuration nk , its Levi
graph G(C) is a k–regular bipartite graph whose vertex set are the points and the
lines of C and there is an edge between a point and a line in the graph if and only
if they are incident in C. We will indistinctly refer to Levi graphs of configurations
as their incidence graphs.
It follows from Theorem 5.9 that an essentially 4–edge–connected pseudo 2–
factor isomorphic cubic bipartite graph of girth greater than or equal to 6 is the
Levi graph of a symmetric configuration n3 .
8
Marien Abreu and Domenico Labbate
In 1886 V. Martinetti [33] characterized symmetric configurations n3 , showing
that they can be obtained from an infinite set of so called irreducible configurations, of which he gave a list. Recently, Boben proved that Martinetti’s list of
irreducible configurations was incomplete and completed it [8]. Boben’s list of irreducible configurations was obtained characterizing their Levi graphs, which he
called irreducible Levi graphs.
In [6] Abreu, Labbate and Sheehan characterized irreducible pseudo 2–factor
isomorphic cubic bipartite graphs (and hence gave a further partial answer to Conjecture 5.5) as follows:
Theorem 5.10 ([6]). The Heawood and the Pappus graphs are the only irreducible
Levi graphs which are pseudo 2–factor isomorphic.
This approach is not feasible to prove Conjecture 5.5 and hence our main Conjecture 3.2 by studying the 2–factors of reducible configurations from the set of
2–factors of their underlying irreducible ones as the following discussion shows.
It is well known that the 73 configuration, whose Levi graph is the Heawood
graph, is not Martinetti extendible and that the Pappus configuration is Martinetti
extendible in a unique way; it is easy to show that this extension is not pseudo
2-factor isomorphic. Let C be a symmetric configuration n3 and C be a symmetric
configuration (n + 1)3 obtained from C through a Martinetti extension. It can be
easily checked that there are 2–factors in C that cannot be reduced to a 2–factor in
C. On the other hand, all of its Martinetti reductions are no longer pseudo 2–factor
isomorphic (for further details cf. [6]).
In the next section we will see that Conjecture 5.5 has been disproved while our
main Conjecture 3.2 still holds.
6. A counterexample to the pseudo 2–factor’s conjecture
In this section we present the counterexample by J. Goedgebeur to the pseudo 2–
factor isomorphic bipartite’s Conjecture 5.5 obtained using an exhaustive research
via parallel computers partially described below (for details refer to [23]).
Using the program minibaum [9], he generated all cubic bipartite graphs with
girth at least 6 up to 40 vertices and all cubic bipartite graphs with girth at least 8
up to 48 vertices. The counts of these graphs can be found in [23, Table 1]. Some of
these graphs can be downloaded from http://hog.grinvin.org/Cubic i.e. the House
of graphs [10]. He then implemented a program which tests if a given graph is
pseudo 2–factor isomorphic and applied it to the generated cubic bipartite graphs.
This yielded the following results:
Remark 6.1 ([23]). There is exactly one essentially 4–edge–connected pseudo 2–
factor isomorphic graph di↵erent from the Heawood graph and the Pappus graph
among the cubic bipartite graphs with girth at least 6 with at most 40 vertices.
Remark 6.2 ([23]). There is no essentially 4–edge–connected pseudo 2–factor isomorphic graph among the cubic bipartite graphs with girth at least 8 with at most
48 vertices.
This implies that Conjecture 5.5 (and consequently also Conjecture 5.6) is false.
The counterexample has 30 vertices and there are no additional counterexamples
up to at least 40 vertices and also no counterexamples among the cubic bipartite
graphs with girth at least 8 up to at least 48 vertices.
2–Factors of regular graphs: an updated survey
9
The counterexample (which we will denote by G) is shown in Figure below. G can
also be obtained from the House of Graphs [10] by searching for the keywords pseudo
2-factor isomorphic *counterexample where it can be downloaded and several of its
invariants can be inspected.
G has cyclic edge–connectivity 6, automorphism group size 144, is not vertex–
transitive, has 312 2–factors and the cycle sizes of its 2–factors are: (6, 6, 18),
(6, 10, 14), (10, 10, 10) and (30). Since all 2–factor hamiltonian graphs are pseudo
2–factor isomorphic and G is not 2–factor hamiltonian, this implies the following:
Remark 6.3 ([23]). Conjecture 3.2 holds up to at least 40 vertices and holds for
cubic bipartite graphs with girth at least 8 up to at least 48 vertices.
7. Strongly pseudo 2-factor isomorphic graphs
The authors and Sheehan in [5] have extended the above mentioned results on
regular pseudo 2–factor isomorphic bipartite graphs to the not necessarily bipartite
case introducing the family of strongly pseudo 2–factor isomorphic graphs:
Definition 7.1. Let G be a graph which has a 2–factor. For each 2–factor F of
G, let t⇤i (F ) be the number of cycles of F of length 2i modulo 4. Set ti to be the
function defined on the set of 2–factors F of G by:
(
0 if t⇤i (F ) is even
ti (F ) =
(i = 0, 1) .
1 if t⇤i (F ) is odd
Then G is said to be strongly pseudo 2–factor isomorphic if both t0 and t1 are
constant functions. Moreover, if in addition t0 = t1 , set t(G) := ti (F ), i = 0, 1.
By definition, if G is strongly pseudo 2–factor isomorphic then G is pseudo 2–
factor isomorphic. On the other hand there exist graphs such as the Dodecahedron
which are pseudo 2–factor isomorphic but not strongly pseudo 2–factor isomorphic:
the 2–factors of the Dodecahedron consist either of a cycle of length 20 or of three
cycles: one of length 10 and the other two of length 5.
In the bipartite case, pseudo 2–factor isomorphic and strongly pseudo 2–factor
isomorphic are equivalent.
10
Marien Abreu and Domenico Labbate
In what follows we will denote by HU , U , SP U and P U the sets of 2–factor
hamiltonian, 2–factor isomorphic, strongly pseudo 2–factor isomorphic and pseudo
2–factor isomorphic graphs, respectively. Similarly, HU (k), U (k), SP U (k), P U (k)
respectively denote the k–regular graphs in HU , U , SP U and P U .
Theorem 7.2 ([5]). Let D be a digraph with n vertices and X be a directed 2–factor
of D. Suppose that either
(a) d+ (v)
+
blog2 nc + 2 for all v 2 V (D), or
(b) d (v) = d (v)
4 for all v 2 V (D).
Then D has a directed 2–factor Y with a di↵erent parity of number of cycles from
X.
Let DSP U and DP U be the sets of digraphs in SP U and P U , i.e. strongly
pseudo and pseudo 2–factor isomorphic digraphs, respectively. Similarly, DSP U (k)
and DP U (k) respectively denote the k–diregular digraphs in DSP U and DP U .
Corollary 7.3 ([5]).
(i) DSP U (k) = DP U (k) = ; for k
4;
(ii) If D 2 DP U then D has a vertex of out–degree at most blog2 nc + 1.
Theorem 7.4 ([5]). Let G be a graph with n vertices and X be a 2–factor of G.
Suppose that either
(a) d(v)
2(blog2 nc + 2) for all v 2 V (G), or
(b) G is a 2k–regular graph for some k
4.
Then G has a 2–factor Y with a di↵erent parity of number of cycles from X.
Corollary 7.5 ([5]).
(i) If G 2 P U then G contains a vertex of degree at most 2blog2 nc + 3;
(ii) P U (2k) = SP U (2k) = ; for k
4.
We know that there are examples of graphs in P U (3), SP U (3), P U (4) and
SP U (4), hence they are not empty and we have seen (cf. Conjecture 4.10) that it
has been conjectured in [1] that HU (4) = {K5 }.
There are many gaps in our knowledge even when we restrict attention to regular
graphs. Some questions arise naturally. Here a few of them.
Problem 7.6. Is P U (2k + 1) = ; for k
2?
In particular we wonder if P U (7) and P U (5) are empty.
Problem 7.7. Is P U (6) empty?
Problem 7.8. Is K5 the only 4–edge–connected graph in P U (4)?
In this paper we have also started to investigate relations between pseudo strongly
2–factor isomorphic graphs and a class of graphs called odd 2–factored snarks. Next
section is devoted to this class of snarks.
2–Factors of regular graphs: an updated survey
11
8. Odd 2–factored snarks
A snark (cf. e.g. [25]) is a bridgeless cubic graph with chromatic index four
(by Vizing’s theorem the chromatic index of every cubic graph is either three or
four so a snark corresponds to the special case of four). In order to avoid trivial
cases, snarks are usually assumed to have girth at least five and not to contain a
non–trivial 3–edge cut (i.e. they are cyclically 4–edge connected).
Snarks were named after the mysterious and elusive creature in Lewis Caroll’s
famous poem The Hunting of The Snark by Martin Gardner in 1976 [20], but it
was P.G. Tait in 1880 that initiated the study of snarks, when he proved that the
four colour theorem is equivalent to the statement that no snark is planar [41]. The
Petersen graph P is the smallest snark and Tutte conjectured that all snarks have
Petersen graph minors. This conjecture was proven by Robertson, Seymour and
Thomas (cf. [37]). Necessarily, snarks are non–hamiltonian.
The importance of the snarks does not only depend on the four colour theorem.
Indeed, there are several important open problems such as the classical cycle double
cover conjecture [38, 39], Fulkerson’s conjecture [16] and Tutte’s 5–flow conjecture
[46] for which it is sufficient to prove them for snarks. Thus, minimal counterexamples to these and other problems must reside, if they exist at all, among the family
of snarks.
At present, there is no uniform theoretical method for studying snarks and their
behaviour. In particular, little is known about the structure of 2–factors in a given
snark.
Snarks play also an important role in characterizing regular graphs with some
conditions imposed on their 2–factors. Recall that a 2–factor is a 2–regular spanning
subgraph of a graph G.
We say that a graph G is odd 2–factored (cf. [5]) if for each 2–factor F of G each
cycle of F is odd.
By definition, an odd 2–factored graph G is pseudo 2–factor isomorphic. Note
that, odd 2–factoredness is not the same as the oddness of a (cubic) graph (cf.
e.g.[47]).
Lemma 8.1 ([5]). Let G be a cubic 3–connected odd 2–factored graph then G is a
snark.
In [5] we have investigated which snarks are odd 2–factored and we have conjectured that:
Conjecture 8.2 ([5]). A snark is odd 2–factored if and only if G is the Petersen
graph, Blanuša 2, or a Flower snark J(t), with t 5 and odd.
In [4], the authors with R. Rizzi and J. Sheehan, present a general construction
of odd 2–factored snarks performing the Isaacs’ dot–product [26] on edges with
particular properties, called bold–edges and gadget–pairs respectively, of two snarks
L and R.
Construction: Bold–Gadget Dot Product. [4]
We construct (new) odd 2–factored snarks as follows:
• Take two snarks L and R with bold–edges (cf. Definition 8.3) and gadget–
pairs (cf. Definition 8.5), respectively;
12
Marien Abreu and Domenico Labbate
• Choose a bold–edge xy in L;
• Choose a gadget–pair f , g in R;
• Perform a dot product L · R using these edges;
• Obtain a new odd 2–factored snark (cf. Theorem 8.7).
Note that in what follows the existence of a 2–factor in a snark is guaranteed
since they are bridgeless by definition.
Definition 8.3 ([4]). Let L be a snark. A bold–edge is an edge e = xy 2 L such
that the following conditions hold:
(i) All 2–factors of L
x and of L
y are odd;
(ii) all 2–factors of L containing xy are odd;
(iii) all 2–factors of L avoiding xy are odd.
Note that not all snarks contain bold–edges (cf. [4, Proposition 4.2] and [4,
Lemma 5.1]). Furthermore, conditions (ii) and (iii) are trivially satisfied if L is
odd 2–factored.
Lemma 8.4 ([4]). The edges of the Petersen graph P10 are all bold–edges.
Definition 8.5 ([4]). Let R be a snark. A pair of independent edges f = ab and
g = cd is called a gadget–pair if the following conditions hold:
(i) There are no 2–factors of R avoiding both f, g;
(ii) all 2–factors of R containing exactly one element of {f, g} are odd;
(iii) all 2–factors of R containing both f and g are odd. Moreover, f and g
belong to di↵erent cycles in each such factor.
(iv) all 2–factors of (R {f, g}) [ {ac, ad, bc, bd} containing exactly one element
of {ac, ad, bc, bd}, are such that the cycle containing the new edge is even
and all other cycles are odd.
Note that, finding gadget–pairs in a snark is not an easy task and, in general,
not all snarks contain gadget–pairs (cf. [4, Lemma 5.2]).
Let H := {x1 y1 , x2 y2 , x3 y3 } be the two horizontal edges and the vertical edge
respectively (in the pentagon–pentagram representation) of P10 (cf. Figure 1).
Lemma 8.6 ([4]). Any pair of distinct edges f, g in the set H of P10 is a gadget–
pair.
The following theorem allows us to construct new odd 2–factored snarks.
Theorem 8.7 ([4]). Let xy be a bold–edge in a snark L and let {ab, cd} be a
gadget–pair in a snark R. Then L · R is an odd 2–factored snark.
In particular, without going into lengthy details (the interested reader might find
those in [4]), this method allows us to construct two instances of odd 2–factored
snarks of order 26 and 34 isomorphic to those obtained by Brinkmann et al. in
[11] through an exhaustive computer search on all snarks of order  36 that has
allowed them to disprove the above conjecture (cf. Conjecture 8.2). Hence, our
construction independently also yields counterexample for Conjecture 8.2.
2–Factors of regular graphs: an updated survey
13
Figure 1. Any pair of the dashed edges is a gadget–pair in P10 .
To approach the problem of characterizing all odd 2–factored snarks, we consider the possibility of constructing further odd 2–factored snarks with the technique presented above, which relies in finding other snarks with bold–edges and/or
gadget–pairs, Therefore, we study the existence of bold–edges and gadget–pairs
in the known odd 2–factored snarks. The results obtained so far give rise to the
following partial characterization:
Theorem 8.8 ([4]). Let G be an odd 2–factored snark of cyclic edge–connectivity
four that can be constructed from the Petersen graph and the Flower snarks using
the bold–gadget dot product construction. Then G 2 {P18 , P26 , P34 }.
Finally, we pose in [4] a new conjecture about odd 2–factored snarks.
Conjecture 8.9 ([4]). Let G be a cyclically 5–edge connected odd 2–factored snark.
Then G is either the Petersen graph or the Flower snark J(t), for odd t 5.
Remark 8.10. (i) A minimal counterexample to Conjecture 8.9 must be a cyclically 5–edge connected snark of order at least 36. Moreover, as highlighted in [11],
order 34 is a turning point for several properties of snarks.
(ii) It is very likely that, if such counterexample exists, it will arise from the superposition operation by M.Kochol [27] applied to one of the known odd 2–factored
snarks.
(iii) J. Goedgebeur [22] checked that none of the snarks (in particular those with
girth 6 of order 38) that G.Brinkmann and himself generates in [12] is an odd
2–factored snark.
References
[1] M. Abreu, R. Aldred, M. Funk, B. Jackson, D. Labbate & J. Sheehan, Graphs and digraphs
with all 2–factor isomorphic, J. Combin. Theory Ser. B, 92(2)(2004), 395–404.
[2] M. Abreu, R. Aldred, M. Funk, B. Jackson, D. Labbate & J. Sheehan, Corrigendum
to “Graphs and digraphs with all 2–factors isomorphic” [J. Combin. Theory Ser B,
92(2)(2004), 395–404], J. Combin. Theory Ser. B, 99(1)(2009), 271–273.
[3] M. Abreu, A. Diwan, B. Jackson, D. Labbate & J. Sheehan, Pseudo 2–Factor isomorphic
regular bipartite graphs, J. Combin. Theory Ser. B, 98(2)(2008), , 432–442.
[4] M. Abreu, D. Labbate, R. RIzzi & J. Sheehan, Odd 2–factored snarks, European J.
Combin., 36(2014), 460–472.
14
Marien Abreu and Domenico Labbate
[5] M. Abreu, D. Labbate & J. Sheehan, Pseudo and strongly pseudo 2–factor isomorphic
regular graphs, European J. Combin., 33(2012), 1847–1856.
[6] M. Abreu, D. Labbate & J. Sheehan, Irreducible pseudo 2–factor isomorphic cubic bipartite graphs, Des. Codes Cryptogr., 64(2012), 153–160.
[7] R. Aldred, M. Funk, B. Jackson, D. Labbate & J. Sheehan, Regular bipartite graphs with
all 2–factors isomorphic, J. Combin. Theory Ser. B, 92(1)(2004), 151–161.
[8] M. Boben, Irreducible (v3 ) configurations and graphs, Discrete Mathematics, 307(2007),
331–344.
[9] G. Brinkmann, Fast generation of cubic graphs, J. Graph Theory, 23(2)(1996), 139–149.
[10] G. Brinkmann, K. Coolsaet, J. Goedgebeur & H. Mlot, House of graphs: a database of
interesting graphs, Discrete Appl. Math., 161(12)(2013), 311--314.
Available at http://hog.grinvin.org/.
[11] G. Brinkmann, J.Goedgebeur, J. Hägglund & K. Markström, Generation and properties
of snarks, J. Combin. Theory Ser. B, 103(4)(2013), 468–488.
[12] G. Brinkmann & J. Goedgebeur, Generation of cubic graphs and snarks with large girth,
Preprint , 2016, ArXiv:1512.05944.
[13] J.A. Bondy & U.S.R. Murty, Graph theory, Graduate Texts in Mathematics, Vol. 244,
Springer Series, 2008.
[14] A.A. Diwan, Disconnected 2–factors in planar cubic bridgeless graphs, J. Combin. Theory
Ser. B, 84(2002), 249–259.
[15] R.J. Faudree, R.J. Gould & M.S. Jacobson, On the extremal number of edges in 2–factor
hamiltonian graphs, Graph Theory - Trends in Mathematics, Birkhäuser, 2006, 139–148.
[16] D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming,
1(1971), 168–194.
[17] M. Funk, B. Jackson, D. Labbate & J. Sheehan, Det-extremal cubic bipartite graphs, J.
Graph Theory, 44(1)(2003), 50–64.
[18] M. Funk, B. Jackson, D. Labbate & J. Sheehan, 2–factor hamiltonian graphs, J. Combin.
Theory Ser. B, 87(1)(2003), 138–144.
[19] M. Funk & D. Labbate, On minimally one–factorable r–regular bipartite graphs, Discrete
Math., 216(2000),121–137.
[20] M. Gardner, Mathematical games: snarks, boojums and other conjectures related to the
four–color–map theorem, Sci. Amer, 234(4)(1976), 126–130.
[21] M. Ghandehari & H. Hatami, A note on independent dominating sets and second hamiltonian cycles, unpublished.
[22] J. Goedgebeur, Private communication, (2015).
[23] J. Goedgebeur, A counterexample to the pseudo 2-factor isomorphic graph conjecture,
Discrete Appl. Math., 193(2015), 57–60.
[24] P. Haxell, B. Seamone & J. Verstraete, Independent dominating sets and Hamiltonian
cycles, J. Graph Theory, 54(3)(2007), 233--244.
[25] D.A. Holton & J. Sheehan, The Petersen graph, Australian Mathematical Society Lecture
Series, 7, Cambridge University Press, Cambridge, 1993.
[26] R. Isaacs, Infinite families on nontrivial trivalent graphs which are not Tait colourable,
Amer. Math. Monthly, 82(1975), 221–239.
[27] M. Kochol, Snarks without small cycles, J. Combin. Theory Ser. B., 67(1)(1996), 34–47.
[28] D. Labbate, On determinants and permanents of minimally 1–factorable cubic bipartite
graphs, Note Mat., 20(1)(2000/01), 37--42.
[29] D. Labbate, On 3–cut reductions of minimally 1–factorable cubic bigraphs, Discrete
Math., 231(2001), 303–310.
[30] D. Labbate, Characterizing minimally 1–factorable r–regular bipartite graphs, Discrete
Math., 248(1-3)(2002), 109–123.
[31] C.H.C. Little, A characterization of convertible (0, 1)–matrices, J. Combin. Theory Ser.
B, 18(1975), 187–208.
[32] L. Lovász & M.D. Plummer, Matching theory, Ann. Discrete Math., 29, North–Holland,
Amsterdam, 1986.
[33] V. Martinetti, Sulle configurazioni piane µ3 , Annali di Matematica Pura ed Applicata,
Ser. II - 15, 1–26, (aprile 1867 - gennaio 1888).
[34] W. McCuaig, Pólya’s permanent problem, Electron. J. Combin., 11(1)(2004), Research
Paper 79, 83 pp.
2–Factors of regular graphs: an updated survey
15
[35] W. McCuaig, Even dicycles, J. Graph Theory, 35(1)(2000), 46--68.
[36] N. Robertson, P. D. Seymour & R. Thomas, Permanents, Pfaffian orientations, and even
directed circuits, Ann. Math. (2), 150(3)(1999), 929--975.
[37] N. Robertson, P. Seymour & R. Thomas, Tutte’s edge-colouring conjecture, J. Combin.
Theory Ser. B, 70(1)(1997), 166–183.
[38] P. D. Seymour, Sums of circuits, Graph Theory and related Topics, Proc. Conf., Univ.
Waterloo, Waterloo, Ont., 1977, Academic Press, New York, 1979, 341–355.
[39] G. Szekeres, Polyhedral decompositions of cubic graphs, Bull. Austral. Math. Soc., 8(1973),
367–387.
[40] J. Sheehan, Problem section, Proc. Fifth British Combinatorial Conf., C.St.J.A. Nash–
Williams & J. Sheehan eds., Congressus Numerantium XV, Utilitas Mathematica Publ.
Corp., Winnipeg, 1975, 691.
[41] P.G. Tait, Remarks on the colourings of maps, Proc. R. Soc. Edinburgh, 10(1880), 729.
[42] A.G. Thomason, Hamiltonian cycles and uniquely 3–edge colourable graphs, Advances in
Graph Theory, B. Bollobás ed., Ann. Discrete Math., 3(1978), 259–268.
[43] C. Thomassen, On the number of hamiltonian cycles in cubic graphs, Combinatorics,
Probability and Computing, 5(1996), 437–442.
[44] C. Thomassen, Independent dominating sets and a second hamiltonian cycle, J. Combin.
Theory Ser. B, 72(1998), 104–109.
[45] W. T. Tutte, On hamiltonian circuits, J. London Math. Soc., 21(1946), 98–101.
[46] W. T. Tutte, A contribution to the theory of chromatic polynomials, Canadian J. Math.,
6(1954), 80–91.
[47] C.-Q. Zhang, Integer flows and cycle covers of graphs, Marcel Dekker Inc., New York,
1997.